Differential equations of elliptic type with variable operators and general Robin boundary condition in UMD spaces

In this paper we study an abstract second order differential equation of elliptic type with variable operator coefficients and general Robin boundary conditions containing an unbounded linear operator. The study is performed when the second member belongs to a Sobolev space and uses the celebrated Dore-Venni theorem. Here, we do not assume the differentiability of the resolvent operators. We give necessary and sufficient conditions on the data to obtain existence, uniqueness of the classical solution satisfying the maximal regularity property are obtained under the Labbas-Terreni assumption. Our techniques use essentially the semigroups theory, fractional powers of linear operators, Dunford's functional calculus and interpolation theory. This work is naturally the continuation of the ones studied by R. Haoua in the UMD spaces and homogenous cases. We also give an example to which our theory applies.